Let us try to extend the analysis to nonlinear systems. Let us consider
stability of the fixed point of a nonlinear system.
Nonlinear system can be linearized by \(f(x)=f(x^*)+f^{\prime }(x^*)(x-x^*)+\ldots \) .
We define \(y=x-x^*\) . Now \(x_{n+1}=f(x_n)\)  is equivalent to
\(y_{n+1}=x_{n+1}-x^*=f(x_n)-x^*=f^{\prime }(x^*)y_n=ry_n\)  where
\(r=f^{\prime }(x^*)\) .
Thus we can conjecture
that the stability regime for the fixed point is
given by \(a_c(\alpha )<f^{\prime }(x^*)<1\) .
Interestingly, this conjecture works.
Consider the cases
of Bernoulli map
and Gauss map
considered by Deshpande and Gejji [1]}.
